A Note on the Security Number of Grid-Like Graphs

Erfang Shan1,2, Hengwu Jiang2
1School of Management, Shanghai University, Shanghai 200444, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Abstract

For any graph \( G = (V, E) \), a non-empty set \( S \subseteq V \) is \({secure}\) if and only if \( |N[X] \cap S| \geq |N[X] – S| \) for all \( X \subseteq S \). The cardinality of a minimum secure set in \( G \) is the security number of \( G \). In this note, we give a new proof for the \({security\; number}\) of grid-like graphs.

Keywords: Security number; Cartesian product; Grid.